Are there any sources that allow/prohibit saying brachot while walking?
Let me start by saying that speaking a Bracha is inherently good. While the Rabbis have preferences for doing a Bracha in an appropriate fashion, you're still coming from an inherently good place if you made a mistake or didn't do something perfectly.
Effort and intention matter just as much as action in this regard. That being said, once you become aware of proper method, you have a responsibility to adhere to the proper method whenever possible.
We are taught that Brachot should be given our complete and full attention. When you are doing a Bracha, you are speaking to Hashem. You shouldn't be distracting yourself with other activities when doing one and you should momentarily stop and reflect on the Bracha with your full attention. - If you were standing before a king, would you be busy with something else while greeting them or would you be giving them your full attention? This concept relates to making a Bracha.
To answer your question:
Can you make a Bracha while walking?
Yes but it isn't considered doing it in best possible form. There is actually a hierarchy.
(Best form) - Sitting during Bracha.- Kaf HaChaim (OC 183:51) (exceptions being when making a Bracha over specific mitzva activities (see Pri Megadim, Introduction to Hilchot Brachot 18)
(Good form) - Standing and making Bracha. (Orach Chaim 183:11, with Mishnah Berurah 36 and Sha’ar HaTziyun 40)
(Acceptable form) If you are walking outside toward a fixed destination, and stopping will make you too anxious to focus on the bracha (i.e. stopping will actually decrease your level of concentration), then you could say the bracha while walking. (Orach Chaim 183:11, with Mishnah Berurah 36)
One thing to be aware of. The third point actually varies from person to person depending on their mental state and anxiety level. What that means is if you actually do for some reason have clearer concentration while walking vs standing still, you would be an exception to the rule. Of course, you'd need to seek guidance from your Rabbi to establish this.
The end goal of the guidance is to ensure that you can make the best possible Bracha with your full concentration. Distraction is the enemy.
I've attached a link to some materials on Brachot protocols that should clarify some things.